# Could gravitational waves be matter waves?

Pretty straightforward. The idea of 3 solar masses radiating away as gravitational waves is still troubling to me. I don't understand how matter can be converted into 'gravitational energy'. I can understand the energy causing the ripples in spacetime, but not the process of matter dissipating into the spacetime continuum (that's probably a butchered account of what really happens). Anyway, the question is: can gravitational waves be explained as matter waves and not as ripples in the spacetime continuum?

• "I don't understand how matter can be converted into 'gravitational energy'." - I assume you're talking about the binary black hole merger where it is reported that the final black hole mass is 3 solar masses less than the sum of the two in-spiraling black hole masses? If so, keep in mind that black holes are not, AFAIK, matter so there was no matter converted into gravitational energy. It's complicated but, for example, there is no matter anywhere, anywhen in the Schwarzschild black hole solution. – Alfred Centauri Jun 17 '17 at 23:01
• Thanks, but I don't think anyone actually knows what black holes are made of. – Sam Cottle Jun 18 '17 at 19:28
• Sam, a Schwarzschild black hole is 'made of' spacetime curvature. – Alfred Centauri Jun 18 '17 at 22:25
• In theory perhaps, but think about the reality, they're formed of collapsing stars so they must be made of some kind of matter (or energy, I don't know). – Sam Cottle Jun 18 '17 at 22:27
• Sam, a Schwarzschild black hole is not formed by collapsing matter; as I wrote above, there is no matter anywhere, anywhen in the Schwarzschild black hole solution. – Alfred Centauri Jun 18 '17 at 22:33

We may categorise waves based on the medium they use to travel; as you are aware, electromagnetic radiation or light does not require a medium and may travel through vacuum, whereas sound waves would correspond to propagating displacements and pressure in matter.

On the other hand, a gravitational wave is precisely a wave which propagates through space-time, in the sense it is some propagating distortion of the metric tensor, $g_{\mu\nu}$.

Gravitational wave solutions exist in the absence of any matter, since they can be shown to satisfy the vacuum Einstein field equations,

$$G_{\mu\nu} = 0$$

corresponding to a vanishing stress-energy tensor. One cannot think of a gravitational wave as a matter wave, say, physically displacing matter. That being said, gravitational waves do have energy, momentum and angular momentum.

Mathematical Details

To go into mathematical detail, the notion of a plane wave in vacuum also exists for gravity, namely one may have a gravitational wave of the form,

$$h_{\mu\nu} = A_{\mu\nu} e^{ik \cdot x} = (h_{+}\epsilon^{+}_{\mu\nu} + h_{\times}\epsilon^{\times}_{\mu\nu})e^{ik\cdot x}$$

with two possible polarisations. In the transverse-trace free gauge, one can think of a gravitational wave as a propagating distortion in curvature as,

$$R_{\mu 0\nu 0} = -\frac12 \partial_th_{\mu\nu}.$$

Of course, gravitational waves may also propagate due to a source $T^{\mu\nu}$, and an approximation of the form of the gravitational wave due to Einstein is given by,

$$h^{\mu\nu} = -\frac{\kappa}{4\pi}\int_V d^3 x' \frac{T^{\mu\nu}(t-|x - x'|, x')}{|x-x'|}$$

which is analogous to how one would evaluate the amplitude for electromagnetism, given a quadrupole moment. We do not think of either physically displacing matter, though both are thought of in QFT as also being represented by gravitons or photons.

In that sense, we may think of a gravitational wave as a matter wave in the QFT sense of it being represented by propagating gauge bosons, namely gravitons.

• But don't they physically displace matter anyway? I mean, they make the arms at LIGO elongate and cause the interference pattern. I was thinking that if they're 'made' of anything they'd be made of some strange form of matter i.e. whatever black holes are made of. Could you expand? – Sam Cottle Jun 17 '17 at 22:55
• @SamCottle, see the comment I just made to your post. Black holes are not matter AFAIK – Alfred Centauri Jun 17 '17 at 23:02
• @SamCottle If you think of a gravitational wave as a graviton, in the same way we think of light as photons, then you could argue they do at least interact with matter, since one can couple the Einstein-Hilbert action to other fields, with interaction terms. – JamalS Jun 17 '17 at 23:03
• @SamCottle Moreover, "matter" changes the propagation geometry of g waves, so it certainly does interact with matter. But I sense that you need the simple insight of Feynman's "sticky bead" argument. This shows the other half of the interaction. – WetSavannaAnimal Jun 18 '17 at 0:06

The energy being radiated away by gravitational waves is provided by the orbital energy (kinetic + gravitational potential) of the merging bodies. Conceptually, it is basically the same as the electromagnetic radiation emitted by accelerating charges - which I assume you don't have a problem with, as that's the basis of how a radio works.

You may be getting confused by the fact that if you lock a system in a closed box, all of the energy contained in the system (mass energy, kinetic energy, gravitational or electromagnetic energy, etc) will contribute to the apparent mass of the box. In that sense, the black hole system will be "lighter" after the merger than it was before.

• Uhm. The merged system has less mass than the sum of the masses of the two initial bodies, and the energy comes ultimately from the gravitational interaction. Certainly part of it is converted to kinetic energy as they spiral in, but you can think of that as a viral effect. – dmckee Jun 18 '17 at 1:04
• @dmckee I suppose I phrased that answer rather poorly, but the point that I was trying to make was that the radiation of gravitational waves is not a conversion of "matter to energy" a la the annihilation of particles and antiparticles, and is much more akin to the radiation of EM waves from accelerating charges. My comment regarding kinetic energy should have included gravitational potential energy as well - edited. – J. Murray Jun 18 '17 at 1:36
• Yes, that is a good point. – dmckee Jun 18 '17 at 1:37

Mass is never "converted into energy". This is just a loose and somewhat misleading way of describing certain nuclear reactions. Here's the mass-energy equivalence relation

$E = mc^2$

(here I'm referring to the total energy/mass). This doesn't say that mass and energy can be converted into one another. It says that mass and energy are the same thing. If you take a jar of water and heat it up a bit, its energy increases, and therefore its mass increases too, because that energy has mass.

The most visceral example of this that I'm aware of is the mass of the helium nucleus. Helium nuclei are composed of two neutrons (m=1.008u) and two protons (m=1.007u). The total mass of the helium nucleus is not 2*1.008 + 2*1.007 u, however, but about 0.03u more than this. The extra mass is due to the potential energy of that the various particles exert upon one another.

The black holes in a binary are moving around very quickly, and are bound to one another gravitationally. That motion (the binding) has kinetic (potential) energy, and that energy has mass. I mean this in the literal sense, that a black hole binary with quickly moving or widely separated partners is more difficult to accelerate than in the opposite case. As the holes spiral in, the total energy goes down*, and so the mass increases. The energy loss is accounted for by gravitational radiation. The radiated waves also have mass: a square of spacetime with waves in it will attract gravitationally like a massive object, and the mass/energy of a system that absorbs the waves will also increase.

When people say that energy is "converted into mass", they mean that energy is used to produce particles that have a rest mass. Some particles have the special property that they continue to have mass, even when not moving and not bound to other particles.

*The kinetic energy actually goes up until the final merger, but the loss of gravitational potential more than makes up for this.

• while it is true energy and mass are different phases of the ''same thing'' it is still technically true that you can interchange between the two cases, regardless of it being the same phenomenon of a two sided coin. It's just that energy is a diffused form of matter and matter is a concentrated form of energy. – Gareth Meredith Jun 7 '18 at 2:35